{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Unit 7: Complex Variables " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Chapt er 36: Applications" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 32 "Section 36.7: the Joukowski map" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Copyright" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Co pyright * 2001 by Addison Wesley Longman, Inc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All rights reserved. No part of this publication may be reproduced, stored in a retrieval sys tem, or transmitted, in any form or by any means, electronic, mechanic al, photocopying, recording, or otherwise, without the prior written p ermission of the publisher. Printed in the United States of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots):\nwith(plottoo ls):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "The Forward Map" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 257 9 "Joukowski" }{TEXT -1 1 " " }{TEXT 258 3 "map" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w=f(z)" "6#/%\"wG-%\"fG6#%\"zG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "z+1/z" "6#,&%\"zG\"\"\"*&F%F%F$!\"\"F%" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f := z -> z+1/z;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "is of special interest in the application of complex var iables to plane fluid flows. The real and imaginary parts of" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(z)=u+i*v" "6#/-%\"fG6#%\"zG,&%\"uG\"\"\"*&%\"iGF*%\"v GF*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "are respectively " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(x, y)= x+x/(x^2 +y^2)" "6#/-%\"uG6$%\"xG%\"yG,&F'\"\"\"*&F'F*,&*$F'\"\"#F**$F(F.F*!\" \"F*" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v(x,y)=y-y/(x^2+y^2)" "6#/-%\"vG6$%\"xG%\"yG,&F(\"\"\"*&F(F*,&*$F' \"\"#F**$F(F.F*!\"\"F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 " that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "U := unapply(evalc(Re(f(x+I*y))),x,y);\nV := unapply( evalc(Im(f(x+I*y))),x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "w" "6#% \"wG" }{TEXT -1 16 "-plane image of " }{XPPEDIT 18 0 "z=exp(i*theta)" "6#/%\"zG-%$expG6#*&%\"iG\"\"\"%&thetaGF*" }{TEXT -1 25 ", the unit ci rcle in the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 11 "-plane, is " }{XPPEDIT 18 0 "w=2*cos(theta)" "6#/%\"wG*&\"\"#\"\"\"-%$cosG6#%&theta GF'" }{TEXT -1 57 ", which we can see by the following Maple calculati on of " }{XPPEDIT 18 0 "f(exp(i*theta))" "6#-%\"fG6#-%$expG6#*&%\"iG\" \"\"%&thetaGF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "w = simplify(evalc(f(exp(I*t heta))));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "As " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 29 " varies through any range of " }{XPPEDIT 18 0 "2*Pi" "6#* &\"\"#\"\"\"%#PiGF%" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "w" "6#%\"wG" } {TEXT -1 16 " varies between " }{XPPEDIT 18 0 "-2" "6#,$\"\"#!\"\"" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "2" "6#\"\"#" }{TEXT -1 33 ". Hence , the unit circle in the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 27 " -plane maps to the segment " }{XPPEDIT 18 0 "-2<=u" "6#1,$\"\"#!\"\"% \"uG" }{XPPEDIT 18 0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 8 " in the " } {XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 32 "-plane. The unit circle in \+ the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 29 "-plane becomes a slit in the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 65 "-plane. Figure 3 6.26 shows what the map does to the rest of the " }{XPPEDIT 18 0 "z" " 6#%\"zG" }{TEXT -1 7 "-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "To construct Figure 36.26, we need the Ma ple function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F := transform((x,y)->[U(x,y),V(x,y)]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "which transforms plot data-structures in the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 39 "-plane, to plot data-structures in th e " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 8 "-plane. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1235 "for \+ k from 1 to 6 do L||k:=NULL;od:\nfor k from -10 to 10 do\nfor j from - 10 to 10 do\np:=.3*k+.3*j*I;\nif Im(p)>0 then L5:=L5,p elif Im(p)<0 th en L6:=L6,p;fi;\nif abs(p)>=1.2 and Im(p)>0 then L1:=L1,p;\nelif abs(p )<.8 and Im(p)>0 then L2:=L2,p;\nelif abs(p)<.8 and Im(p)<0 then L3:=L 3,p;\nelif abs(p)>=1.2 and Im(p)<0 then L4:=L4,p;fi;\nod:od:\npp1 := c omplexplot([L1],style=point, symbol=cross, color=cyan):\npp2 := comple xplot([L2],style=point, symbol=POINT, color=black):\npp3 := complexplo t([L3],style=point, symbol=cross, color=cyan):\npp4 := complexplot([L4 ],style=point, symbol=POINT, color=black):\npp5 := circle([0,0],1,colo r=red,thickness=3):\npp6 := complexplot([L5],style=point, symbol=cross , color=cyan):\npp7 := complexplot([L6],style=point, symbol=POINT, col or=black):\npp8 := line([-2,0],[2,0],color=red,thickness=3):\npp9 := s ubs(AXESLABELS(\"\",\"\",DEFAULT)=AXESLABELS(\"\",\"\"), display([pp|| (6..8)],scaling=constrained, title=\"w-plane\")):\npp10 := subs(AXESLA BELS(\"\",\"\",DEFAULT)=AXESLABELS(\"\",\"\"), display([pp||(1..5)],sc aling=constrained, title=\"z-plane\")):\npp11 := display(array([pp10,p p9]),xtickmarks=[-3,-2,2,3], ytickmarks=[-3,3]):\npp12 := textplot(\{[ 0,-12,`z-plane`],[30,-12,`w-plane`]\}, font=[TIMES,ROMAN,10]):\ndispla y([pp11,pp12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "(The extra coding for two compone nts of Figure 36.26 circumvent a display/array bug introduced in Maple 6.)" }}{PARA 0 "" 0 "" {TEXT -1 38 "\nPoints are selected on a grid i n the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 87 "-plane. Their loca tion is determined, and on that basis, assigned a color. Then, the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 63 "-plane graph of these point s is transformed, or mapped, to its " }{XPPEDIT 18 0 "w" "6#%\"wG" } {TEXT -1 21 "-plane counterpart. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "The interior of the unit circle in the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 21 "-plane maps onto the " } {XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 73 "-plane. The upper-half of t he unit disk maps onto the lower-half of the " }{XPPEDIT 18 0 "w" "6#% \"wG" }{TEXT -1 78 "-plane, while the lower-half of the unit disk maps onto the upper-half of the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 44 "-plane. In addition, the upper-half of the " }{XPPEDIT 18 0 "z" " 6#%\"zG" }{TEXT -1 67 "-plane exterior to the unit circle maps onto th e upper-half of the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 34 "-plan e, and the lower-half of the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 67 "-plane exterior to the unit circle maps onto the lower-half of the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 95 "-plane. Thus, the Jouko wski map is not one-to-one, and the inverse mapping is multiple-valued ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "The \+ following animation shows that points outside the unit circle and in t he upper-half of the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 43 "-pla ne map to all of the upper-half of the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 147 "-plane. The animation shows the image of a semicircle \+ of increasing radius, starting with a radius of 1. Hence, the first i mage is just the slit " }{XPPEDIT 18 0 "-2<=u" "6#1,$\"\"#!\"\"%\"uG" }{XPPEDIT 18 0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 8 " in the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 7 "-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "p1 := t -> arc([0,0], t,0..Pi,color=red,thickness=3):\ndisplay([seq(F(p1(k/2)),k=2..10)], in sequence=true, scaling=constrained, labels=[u,v], labelfont=[TIMES,ITA LIC,12], xtickmarks=5, ytickmarks=5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "The lower-hal f of the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 68 "-plane outside t he unit circle maps to all of the lower-half of the " }{XPPEDIT 18 0 " w" "6#%\"wG" }{TEXT -1 48 "-plane, as suggested by the following anima tion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "p2 := t -> arc([0,0],t,Pi..2*Pi,color=green,thicknes s=3):\ndisplay([seq(F(p2(k/2)),k=2..10)], insequence=true, scaling=con strained, labels=[u,`v `], labelfont=[TIMES,ITALIC,12], xtickmarks=5, ytickmarks=5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "The upper-half of the interior of \+ the unit circle maps to the lower-half of the " }{XPPEDIT 18 0 "w" "6# %\"wG" }{TEXT -1 48 "-plane, as suggested by the following animation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "p3 := t -> arc([0,0],t,0..Pi,color=blue,thickness=3) :\ndisplay([seq(F(p3(k/10)),k=2..10)], insequence=true, scaling=constr ained, labels=[u,v], labelfont=[TIMES,ITALIC,12], xtickmarks=5, ytickm arks=5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Finally, the lower-half of the " } {XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 65 "-plane interior to the unit \+ circle maps to the upper-half of the " }{XPPEDIT 18 0 "w" "6#%\"wG" } {TEXT -1 48 "-plane, as suggested by the following animation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "p4 := t -> arc([0,0],t,Pi..2*Pi,color=black,thickness=3):\ndisp lay([seq(F(p4(k/10)),k=2..10)], insequence=true, scaling=constrained, \+ labels=[u,v], labelfont=[TIMES,ITALIC,12], xtickmarks=5, ytickmarks=5) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Thus, the upper-half of the " }{XPPEDIT 18 0 "z" "6# %\"zG" }{TEXT -1 62 "-plane exterior to the unit circle, and the lower -half of the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 70 "-plane inter ior to the unit circle, both map to the upper-half of the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 121 "-plane. A more static graphic for t his conclusion is provided by the following figures. First, a represe ntation of the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 103 "-plane is constructed, with the \"upper-exterior\" and the \"lower-interior\" a ppropriately differentiated." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 725 "qf := NULL:\nqp := NULL:\nf or k from -40 to 40 do\nfor j from -10 to 0 do\np := .1*k+.1*j*I;\nif \+ p<>0 and abs(p)<=1 then qf:=qf,f(p);qp:=qp,p;fi;\nod:\nod:\np5 := comp lexplot([qp],style=point, symbol=cross, scaling=constrained, color=gre en):\np6 := complexplot([qf],style=point, symbol=cross, scaling=constr ained, color=green):\nqf := NULL:\nqp := NULL:\nfor k from -20 to 20 d o\nfor j from 0 to 20 do\np := .1*k+.1*j*I;\nif evalf(abs(p)>1) then q f:=qf,f(p);qp:=qp,p;fi;\nod:\nod:\np7 := complexplot([qp],style=point, symbol=circle, scaling=constrained, color=red):\np8 := complexplot([q f],style=point, symbol=circle, scaling=constrained, color=red):\ndispl ay([p5,p7], labels=[x,y], xtickmarks=4, ytickmarks=[-1,0,2], labelfont =[TIMES,BOLD,14]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The image of that portion of the \+ " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 137 "-plane depicted in this \+ figure is given in the following figure. The red circles from the upp er half-plane map to the upper-half of the " }{XPPEDIT 18 0 "w" "6#%\" wG" }{TEXT -1 91 "-plane. As well, the green crosses from the lower h alf-plane map to the upper-half of the " }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 7 "-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "display([p6,p8],view=[-5..5,0..6], label s=[u,v], labelfont=[TIMES,BOLD,14], xtickmarks=4, ytickmarks=[0,2,4,6] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 260 "Rather than repeat the same demonstration for the \"lower-exterior\" and the \"upper-interior,\" we provide Figure 36.2 6 which shows the both pairs of regions and their images under the Jou kowski map. Although presented above, it is repeated here for conveni ence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1235 "for k from 1 to 6 do L||k:=NULL;od:\nfor k from -10 to 10 do\nfor j from -10 to 10 do\np:=.3*k+.3*j*I;\nif Im(p)>0 then L 5:=L5,p elif Im(p)<0 then L6:=L6,p;fi;\nif abs(p)>=1.2 and Im(p)>0 the n L1:=L1,p;\nelif abs(p)<.8 and Im(p)>0 then L2:=L2,p;\nelif abs(p)<.8 and Im(p)<0 then L3:=L3,p;\nelif abs(p)>=1.2 and Im(p)<0 then L4:=L4, p;fi;\nod:od:\npp1 := complexplot([L1],style=point, symbol=cross, colo r=cyan):\npp2 := complexplot([L2],style=point, symbol=POINT, color=bla ck):\npp3 := complexplot([L3],style=point, symbol=cross, color=cyan): \npp4 := complexplot([L4],style=point, symbol=POINT, color=black):\npp 5 := circle([0,0],1,color=red,thickness=3):\npp6 := complexplot([L5],s tyle=point, symbol=cross, color=cyan):\npp7 := complexplot([L6],style= point, symbol=POINT, color=black):\npp8 := line([-2,0],[2,0],color=red ,thickness=3):\npp9 := subs(AXESLABELS(\"\",\"\",DEFAULT)=AXESLABELS( \"\",\"\"), display([pp||(6..8)],scaling=constrained, title=\"w-plane \")):\npp10 := subs(AXESLABELS(\"\",\"\",DEFAULT)=AXESLABELS(\"\",\"\" ), display([pp||(1..5)],scaling=constrained, title=\"z-plane\")):\npp1 1 := display(array([pp10,pp9]),xtickmarks=[-3,-2,2,3], ytickmarks=[-3, 3]):\npp12 := textplot(\{[0,-12,`z-plane`],[30,-12,`w-plane`]\}, font= [TIMES,ROMAN,10]):\ndisplay([pp11,pp12]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 202 "The inverse map is examined in the last part of this section. \+ Prior to that, however, we examine Joukowski airfoils, the reason why the Joukowski map appears so often in the literature of planar flows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "The Joukows ki Airfoil" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The Joukowski map takes circles which contain " } {XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"" }{TEXT -1 44 " in their interiors , and which pass through " }{XPPEDIT 18 0 "z=-1" "6#/%\"zG,$\"\"\"!\" \"" }{TEXT -1 23 ", to \"airfoils\" in the " }{XPPEDIT 18 0 "w" "6#%\" wG" }{TEXT -1 60 "-plane. Figure 36.27 shows on the left, the circle \+ through " }{XPPEDIT 18 0 "z=-1" "6#/%\"zG,$\"\"\"!\"\"" }{TEXT -1 14 " , with center " }{XPPEDIT 18 0 "``(1/5,1/5)" "6#-%!G6$*&\"\"\"F'\"\"&! \"\"*&F'F'F(F)" }{TEXT -1 12 " and radius " }{XPPEDIT 18 0 "sqrt(17)/5 " "6#*&-%%sqrtG6#\"#<\"\"\"\"\"&!\"\"" }{TEXT -1 84 ". On the right i n Figure 36.27 is the image of this circle under the Joukowski map." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 411 "p9 := t -> circle([t,t],sqrt(2*t^2+2*t+1),color=black):\nR1 := \+ rectangle([-1.5,3.5],[3.5,-1.5],color=white):\nR2 := rectangle([-2,3.2 ],[3.6,-1],color=white):\nfor k from 1 to 10 do\nqq := p9(.1*k):\npc|| k := display([qq,R1],xtickmarks=[-2,-1,1,2,3], ytickmarks=[-1,1,2,3]): \npj||k := display(array([pc||k,display([F(qq),R2])]), xtickmarks=[-2, -1,1,2,3], ytickmarks=[-1,1,2,3]);\nod:\ndisplay([pj2],scaling=constra ined);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The following animation shows, on the left, cir cles in the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 47 "-plane, and, \+ on the right, their images in the " }{XPPEDIT 18 0 "w" "6#%\"wG" } {TEXT -1 7 "-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "p9 := t -> circle([t,t],sqrt(2*t^2+2*t+1 ),color=black):\nR1 := rectangle([-1.5,3.5],[3.5,-1.5],color=white):\n R2 := rectangle([-2,3.2],[3.6,-1],color=white):\nfor k from 1 to 10 do \nqq := p9(.1*k):\npc||k := display([qq,R1],xtickmarks=[-2,-1,1,2,3], \+ ytickmarks=[-1,1,2,3]):\npj||k := display(array([pc||k,display([F(qq), R2])]), xtickmarks=[-2,-1,1,2,3], ytickmarks=[-1,1,2,3]);\nod:\ndispla y([pj||(1..10)],insequence=true,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The circles shown have centers at " }{XPPEDIT 18 0 "``(t,t)" "6 #-%!G6$%\"tGF&" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "1/10<=t" "6#1*& \"\"\"F%\"#5!\"\"%\"tG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 31 ", and have corresponding radii " }{XPPEDIT 18 0 "r=sqrt((t-1)^2 +t^2)" "6#/%\"rG-%%sqrtG6#,&*$,&%\"tG\"\"\"F,!\"\"\"\"#F,*$F+F.F," } {TEXT -1 60 ". These circles satisfy the constraints of passing throu gh " }{XPPEDIT 18 0 "z=-1" "6#/%\"zG,$\"\"\"!\"\"" }{TEXT -1 16 " and \+ containing " }{XPPEDIT 18 0 "z=1" "6#/%\"zG\"\"\"" }{TEXT -1 20 " in t heir interiors." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "A more general version of the Joukowski airfoil is possib le if the Joukowski map is defined by" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(z)=z+k^2/z" "6# /-%\"fG6#%\"zG,&F'\"\"\"*&%\"kG\"\"#F'!\"\"F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "k* `>`*0" "6#*(%\"kG\"\"\"%\">GF%\"\"!F%" }{TEXT -1 41 ". In this case, \+ circles passing through " }{XPPEDIT 18 0 "z=-k" "6#/%\"zG,$%\"kG!\"\" " }{TEXT -1 16 " and containing " }{XPPEDIT 18 0 "z=k" "6#/%\"zG%\"kG " }{TEXT -1 53 " in their interiors are mapped to Joukowski airfoils. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "If th e circle passes through " }{XPPEDIT 18 0 "z=k" "6#/%\"zG%\"kG" }{TEXT -1 14 " and contains " }{XPPEDIT 18 0 "z=-k" "6#/%\"zG,$%\"kG!\"\"" } {TEXT -1 66 " in its interior, the airfoil will face in the opposite d irection." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "The Inverse Map" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "The inverse of the Joukowski m ap is obtained by solving the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "w=f(z)" "6#/%\"wG -%\"fG6#%\"zG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "z+1/z" "6#,&%\"zG\"\" \"*&F%F%F$!\"\"F%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 42 ". Since this map is two-to-one, we obtain" }}{PARA 258 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "z=1/2" "6#/%\"zG*&\"\"\"F&\"\"#!\"\" " }{TEXT -1 2 " (" }{XPPEDIT 18 0 "w" "6#%\"wG" }{TEXT -1 1 " " } {TEXT 259 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(w^2-4)" "6#-%%sqr tG6#,&*$%\"wG\"\"#\"\"\"\"\"%!\"\"" }{TEXT -1 3 " ) " }}{PARA 0 "" 0 " " {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solve(w=f(z),z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "a nd define the inverse by" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(w) = 1/2" "6#/-%\"gG6#%\"wG*&\"\"\"F)\"\"#!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "``(w+sqrt(w^2-4))" "6#-%!G6#,&%\"wG\"\" \"-%%sqrtG6#,&*$F'\"\"#F(\"\"%!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "with the square root d efined so that its discontinuity is along the line segment " } {XPPEDIT 18 0 "-2<=u" "6#1,$\"\"#!\"\"%\"uG" }{XPPEDIT 18 0 "``<=2" "6 #1%!G\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v=0" "6#/%\"vG\"\"!" } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "w=u+i*v" "6#/%\"wG,&%\"uG\"\"\"* &%\"iGF'%\"vGF'F'" }{TEXT -1 101 ". The principal square root does no t satisfy this constraint, as shown in Figure 26.28 where, under " } {XPPEDIT 18 0 "z=g(w)" "6#/%\"zG-%\"gG6#%\"wG" }{TEXT -1 36 ", the inv erse images of the circles " }{XPPEDIT 18 0 "abs(w-2)=1" "6#/-%$absG6# ,&%\"wG\"\"\"\"\"#!\"\"F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "abs(w-2 )=5" "6#/-%$absG6#,&%\"wG\"\"\"\"\"#!\"\"\"\"&" }{TEXT -1 79 " can be \+ seen. The image of the smaller circle (which cuts through the segment " }{XPPEDIT 18 0 "-2<=u" "6#1,$\"\"#!\"\"%\"uG" }{XPPEDIT 18 0 "``<=2 " "6#1%!G\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v=0" "6#/%\"vG\"\"!" }{TEXT -1 101 ") is on the left, and it is continuous. The image of t he larger circle (which does not pass between " }{XPPEDIT 18 0 "u" "6# %\"uG" }{TEXT -1 3 " = " }{TEXT 260 1 "+" }{TEXT -1 4 " 2, " } {XPPEDIT 18 0 "v=0" "6#/%\"vG\"\"!" }{TEXT -1 43 ") is on the right, a nd it is discontinuous!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "To construct Figure 36.28, we define " }{XPPEDIT 18 0 "g(w)" "6#-%\"gG6#%\"wG" }{TEXT -1 3 " as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "G := w -> (w+sqrt (w^2-1))/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "where, by default, Maple will use the pr incipal square root. Using this function, we obtain Figure 36.28 by e xecuting the following Maple commands." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 538 "p10 := complexplot(G(2 +exp(I*t)),t=-Pi..Pi,view=[0..2.7,-1.5..1.5], color=cyan, thickness=3, xtickmarks=[0,1,2], ytickmarks=[-1,1]):\np11 := complexplot(G(2+5*exp (I*t)),t=-Pi..Pi, style=point, numpoints=300, color=black,symbol=diamo nd, xtickmarks=[-2,2,4,6], ytickmarks=[-2,2]):\np12 := textplot(\{[1.5 ,.25,`x`],[.1,1.4,`y`]\}):\np13 := textplot(\{[7.6,.4,`x`],[.15,5.5,`y `]\}):\np14 := display([p10,p12],title=`g applied to |w - 2| = 1`):\np 15 := display([p11,p13],title=`g applied to |w - 2| = 5`):\ndisplay(ar ray([p14,p15]),scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "The appropria te definition of the square root is" }}{PARA 258 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt(w^2-4)=sqrt(abs(w^2-4))*exp(i*[Arg(w-2)+Arg(w+2 )]/2)" "6#/-%%sqrtG6#,&*$%\"wG\"\"#\"\"\"\"\"%!\"\"*&-F%6#-%$absG6#,&* $F)F*F+F,F-F+-%$expG6#*(%\"iGF+7#,&-%$ArgG6#,&F)F+F*F-F+-F>6#,&F)F+F*F +F+F+F*F-F+" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "Arg(w)" "6#-%$ArgG6#%\"wG" }{TEXT -1 37 " is the principal argument, so that " }{XPPEDIT 18 0 "-Pi " 0 "" {MPLTEXT 1 0 267 "p16 := plot([[-2,0],[3,2],[2,0]],color=black):\np17 \+ := textplot(\{[-1.3,.14,`a`],[2.25,.15,`b`]\}, font=[SYMBOL,10]):\np18 := textplot(\{[3.1,2,`w`],[2.8,1,`w - 2`], [1.1,1.5,`w + 2`]\}):\ndis play([p||(16..18)],scaling=constrained, xtickmarks=[-2,0,2], ytickmark s=[-2,0,3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "This branch of the square root is relativ ely easy to construct in Maple by using the " }{TEXT 256 8 "argument" }{TEXT -1 37 " function, Maple's implementation of " }{XPPEDIT 18 0 "A rg(eta)" "6#-%$ArgG6#%$etaG" }{TEXT -1 53 ", the principal argument fu nction. We therefore have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "g := w -> (w+sqrt(abs(w^2-4))*exp(I *(argument(w-2)+argument(w+2))/2))/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "That this bra nch of the inverse function has a branch cut along " }{XPPEDIT 18 0 "- 2<=u" "6#1,$\"\"#!\"\"%\"uG" }{XPPEDIT 18 0 "``<=2,v=0" "6$1%!G\"\"#/% \"vG\"\"!" }{TEXT -1 55 ", is seen from the following figure which sho ws in the " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 51 "-plane, the pre -images of points along the circles " }{XPPEDIT 18 0 "w=2+r*exp(i*thet a),r=1,5" "6%/%\"wG,&\"\"#\"\"\"*&%\"rGF'-%$expG6#*&%\"iGF'%&thetaGF'F 'F'/F)F'\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 514 "p19 := complexplot(g(2+exp( I*t)),t=-Pi..Pi,view=[0..2.7,-1.5..1.5], style=point, numpoints=300, c olor=black,symbol=diamond, xtickmarks=[0,1,2], ytickmarks=[-1,1]):\np2 0 := complexplot(g(2+5*exp(I*t)),t=-Pi..Pi,color=black, thickness=3, x tickmarks=[-2,2,4,6], ytickmarks=[-2,2]):\np21 := textplot(\{[1.5,.25, `x`],[.1,1.4,`y`]\}):\np22 := textplot(\{[7.5,.3,`x`],[.15,5.5,`y`]\}) :\np23 := display([p19,p21], scaling=constrained):\np24 := display([p2 0,p22], scaling=constrained):\ndisplay(array([p23,p24]),scaling=constr ained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 267 "The discontinuous curve on the left is t he pre-image of the circle which has radius 1 and which passes through the branch cut. The continuous curve on the right is the pre-image o f the circle which has radius 5 and which encloses, but does not cross , the branch cut." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Finally, Figure 36.30 shows in the " }{XPPEDIT 18 0 "z" " 6#%\"zG" }{TEXT -1 29 "-plane, the pre-image of the " }{XPPEDIT 18 0 " w" "6#%\"wG" }{TEXT -1 54 "-plane under this branch of the inverse Jou kowski map." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "complexplot([seq(seq(evalf(g(k/10+j/10*I)),k=-40. .40),j=-40..40)], style=point, symbol=POINT, color=black, labels=[x,y] , labelfont=[TIMES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "This figure is consist ent with the conclusions drawn for the (forward) Joukowski map, " } {XPPEDIT 18 0 "w=f(z)" "6#/%\"wG-%\"fG6#%\"zG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "z+1/z" "6#,&%\"zG\"\"\"*&F%F%F$!\"\"F%" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }